10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_929_584_237_287}
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\caption{Figure 4}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_716_544_452_1069}
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\caption{Figure 5}
\end{figure}
Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house.
The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\).
The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\).
The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).
- Show that
$$y = \frac { 320 } { x ^ { 2 } }$$
- Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by
$$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
- Use calculus to find the minimum value of \(S\).
- Justify, by further differentiation, that the value of \(S\) you have found is a minimum.